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Physical Review Letters

Monte Carlo technique for universal finite-size-scaling functions: Application to the 3-state Potts model on a square lattice

Koo-Chul Lee

Published Jul 6, 1992• 29 citations

Abstract

We present a method which allows calculation of whole universal finite-size-scaling functions from Monte Carlo data. The crux of the new method is a technique of isolating the singular part from the total free energy. We apply this method to the 3-state Potts model on a square lattice and find the normalized scaling function Y(x) in the form 1+x+5.31${\mathit{x}}^{2}$-1.2${\mathit{x}}^{3}$-0.67${\mathit{x}}^{4}$ . . . near the bulk critical point x=0, together with the normalizing universal amplitude E=1.053 and a nonuniversal metric factor D=-0.387. It also behaves as Y(x)\ensuremath{\approxeq}(\ifmmode\pm\else\textpm\fi{}x${)}^{\mathit{d}\ensuremath{\nu}}$${\mathit{Q}}_{\ifmmode\pm\else\textpm\fi{}}$ for large x in agreement with the hyperuniversality hypothesis with ${\mathit{Q}}_{+}$/${\mathit{Q}}_{\mathrm{\ensuremath{-}}}$\ensuremath{\approxeq}1.

DOI: 10.1103/physrevlett.69.9

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We present a method which allows calculation of whole universal finite-size-scaling functions from Monte Carlo data. The crux of the new method is a technique of isolating the singular part from the total free energy. We apply this method to the 3-state Potts model on a square lattice and find the normalized scaling function Y(x) in the form 1+x+5.31

{\mathit{x}}^{2}

-1.2

{\mathit{x}}^{3}

-0.67

{\mathit{x}}^{4}

. . . near the bulk critical point x=0, together with the normalizing universal amplitude E=1.053 and a nonuniversal metric factor D=-0.387. It also behaves as Y(x)\ensuremath{\approxeq}(\ifmmode\pm\else\textpm\fi{}x

{)}^{\mathit{d}\ensuremath{\nu}}$${\mathit{Q}}_{\ifmmode\pm\else\textpm\fi{}}

for large x in agreement with the hyperuniversality hypothesis with

{\mathit{Q}}_{+}

/

{\mathit{Q}}_{\mathrm{\ensuremath{-}}}

\ensuremath{\approxeq}1.

The functional Monte Carlo method devised by one of the authors for the efficiency and accuracy can be used to calculate universal scaling functions. In order to demonstrate the technique we take 3-state Potts model for which many of the exact critical properties are known. Various thermodynamic functions of Potts model on square lattice of sizes ranging from 6 X 6 to 33 X 33 are calculated and the two scale factor universality (or hyperuniversality) theory put forward by Privman and Fisher is tested and verified. We also estimate the higher order derivatives of universal function using hyperuniversality hypothesis.

The study of the universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations provides an accurate method of determining the critical temperature and the universality class of phase transitions. These universal amplitudes can replace the critical exponents to identify universality classes. The authors use Bennett's optimisation scheme (1976) for the calculation of the interfacial free energies in Monte Carlo simulations. Tests for the Ising and three-state Potts model confirm the effectiveness of this method.

The idea of universal finite-size-scaling functions of the Ising model is tested by Monte Carlo simulations for various lattices. Not only regular lattices such as the square lattice but quasiperiodic lattices such as the Penrose lattice are treated. We show that the finite-size-scaling functions of the order parameter for various lattices are collapsed on a single curve by choosing two nonuniversal scaling metric factors. We extend the idea of the universal finite-size-scaling functions to the order-parameter distribution function. We pay attention to the effects of boundary conditions.

The idea of universal finite-size-scaling functions of the Ising model is tested by Monte Carlo simulations for various lattices. Not only regular lattices such as the square lattice but quasiperiodic lattices such as the Penrose lattice are treated. We show that the finite-size-scaling functions of the order parameter for various lattices are collapsed on a single curve by choosing two nonuniversal scaling metric factors. We extend the idea of the universal finite-size-scaling functions to the order-parameter distribution function. We pay attention to the effects of boundary conditions.

We derive the exact values of the universal amplitudes associated with the finite-size-scaling behavior of the free energy and interfacial free energies of the Potts model on a torus, by means of the extended-scaling method. These amplitudes vary continuously with the value of the aspect ratio of the torus. Our results make it possible to apply the type of finite-size-scaling techniques that are standard in the context of transfer-matrix calculations to Monte Carlo simulations, and thus enhance the effectiveness of these simulations in establishing the universality class of a phase transition.